# Maxwell equations with time-dependent boundary conditions

1. Aug 4, 2010

### checkfrogger

Hi folks,

I was wondering how to code a Maxwell solver for a problem with time-dependent boundary conditions. This is not my homework, but I love programming and would like to implement what I learned in my physics undergrad course to get a better understanding.
More precisely, if I have an electrode with a time-dependent potential, how do I obtain the electric and magnetic field around it?
I basically came up with two ways, which both seem inappropriate to me:
1)
- calculate the potential using the Poisson equation with boundary conditions at time t=0
- then obtain E as the neg. gradient of the potential.
- calculate E and B at the next time step using the two curl equations of Maxwell equations
- repeat steps 1 and 2 at the next step and it might not be consistent with the third step...
I have the feeling that I mix electrostatics and electrodynamics here
2)
- set a boundary condition for E, solve the divergence equations of Maxwell equations at t=0
- calculate the next time step using the curl equations of Maxwell equations. The obtained E at t=1 might be inconsistent with the new potential at t=1.
-> similar problem here: I am not sure how to make this self-consistent

Andrew

2. Aug 4, 2010

### Born2bwire

Easiest way is the Finite Difference Time Domain (FDTD) analysis using the Yee algorithm which discretizes the differential equations in both time and position.

3. Aug 4, 2010

### checkfrogger

Thanks, but as far as I understand this algorithm only solves the two curl equations. It is fine if you start with a solution that satisfies the divergence equations.
My problem is still how to incorporate boundary conditions which are time-dependent and given in terms of the potential only.

4. Aug 4, 2010

### Born2bwire

Many excitations in computational electromagnetics are given as voltages. The simplest is to do a delta gap source which is simple to do in FDTD as well. More complex methods would involve say finding the principle excitation mode of your source (like in a transmission line) and exciting the principle mode's field. Without having been given what the excitation is we can't really begin to provide any advice on how to model it. Still, any time or frequency domain computational solver like FDTD, finite element method (FEM) or method of moments (MOM) will probably be satisfactory for you. I would suggest looking at an appropriate text to see how excitations are handled. Taflove is good for FDTD and I like a recent text by Walter Gibson for MOM.